Limit Distributions and Random Trees Derived from the Birthday Problem with Unequal Probabilities

By Michael Camarri, Jim Pitman. 2000

111 citations recorded by the source corpus.

Given an arbitrary distribution on a countable set, consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat times are derived by embedding in a Poisson process. In particular, necessary and sufficient conditions for convergence are given and the possible limits explicitly described. Under the same conditions the finite dimensional distributions of the repeat times converge to the arrival times of suitably modified Poisson processes, and random trees derived from the sequence of independent trials converge in distribution to an inhomogeneous continuum random tree.

Subfields: Artificial Intelligence, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/45f86cc64a535279edb0965d37425c5b8fd4aec1

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Citations in the corpus

A Coincidence-Based Test for Uniformity Given Very Sparsely Sampled Discrete Data (2008, background): “…computer science (Batu, 2001; Bellare and Kohno, 2004) and statistics (Diaconis and Mosteller, 1989) literature; in particular, there are obvious connections to the “birthday problem” (Camarri and Pitman, 2000; Das Gupta, 2005) and related techniques for entropy estimation (Nemenman et al., 2004).”
Limit of normalized quadrangulations: The Brownian map (2004, background): “M R1207226 [3] A ldous,D .and Pitman,P.(1999).”
Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent (2000, background): “The material below on discrete trees is developed further in [9], and we give only what is needed for the present paper.”
The matching, birthday and the strong birthday problem: a contemporary review (2005, background): “For other references on the unequal probability case, Camarri and Pitman (2000) is an excellent source.”
Combinatorial Stochastic Processes (2006, background): “[141, 82] Let p = (pj , j ∈ J) be a non-degenerate probability distribution on a finite or infinite set J .”